AD-A109 738 PITTSSURSH UNIV PA INST FOR STATISTICS AND APPLICATIONS F/0 12/1 N

ON THE AXIOMATIC THEORYV OF MULTISTATE COH4ERENT STRUCTURES. (U)I.OCT SI W DE SOU~ZA BRKS, F W RODRIGUES NOOOIVTS5C-0639

UNCLASSIFIED TR-1-1

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MICROCOPY RESOLUIION ltt l HART

LEVEL

ON THE AXIOMATIC THEORY OF

MULTISTATE COHEFZ.NT STRUCTURES

by

Wagner de Souza Borges*t

and

Flvio Wagner Rodrigues

DTICS JAN 1 9 1982

October 1981 BTechnical Report No. 81-33

Institute for Statistics and ApplicationsDepartment of Mathematics and Statistics

University of PittsburghPittsburgh, PA 15260

DISTRIBUTION STATEMENT AKApproved for public refease; Distribution Unlimited

tThe work of this author has been partially supported by ONRContract N00014-76-C-0839.

*

S On leave from the Institute de Matematica e Estatistica,Universidade de Sao Paulo, Sao Paulo, Brasil.

01 1882 017

ON THE AXIOMATIC THEORY OF MULTISTATE

COHERENT STRUCTURES

Wagner de Souza Borges

and

Fl~vio Wagner Rodrigues

Instituto de Matematica e EstatisticaUniversidade de Sio Paulo

Slo Paulo - Brasil

1. ABSTRACT

Mathematical models for multistate reliability systems of

multistate components have been proposed by Barlow & Wu (1978),

El Neweihi et al (1978) and Griffiths (1981). Unlike the

approach used by Barlow & Wu, the other authors preferred to

establish their classes of models through sets of axioms, all

extending the early binary notions and all containing as special

cases the class of models suggested by Barlow & Wu. Since the

Barlow & Wu approach is essentially set theoretic, and since

in the other two approaches these models were not characterized

among the larger classes, one question that arises is whether

these models can be characterized by a set of axioms in the same

way as their counterparts. In this paper we do just that and

obtain a better understanding of Barlow & Wu models.,

AMS Subject Classification: Primary 62N05, Secondary 60K10 (90B25)

Key Words: Multistate System Structures, Reliability.

Currently visiting the Department of Mathematics and Statistics,University of Pittsburgh, Pittsburgh, PA 15260.

2

2. INTRODUCTION, NOTATION AND TEIINOLOGY

A central problem in reliability theory is to determine

the relationship between the reliability of a complex system

and the reliabilities of its components. When the system and

its components are considered to be in either the functioning

(state 1) or the fail (state 0) state, the theory of binary

coherent structures, as developed in Barlow & Proschan (1975),

provides an adequate model that serves as a unifying foundation

for mathematical and statistical aspects of reliability theory.

However, in many practical situations, systems, as well as

their components, can assume a wide range of performance levels,

thus motivating the development of mathematical models which

generalize the binary. Models, representing multistate systems

of multistate components, have been investigated by Barlow & Wu

(1978), El-Neweihi et al C1978) and Griffiths (1981). While

the approach used by Barlow & Wu is based on the minimal path

and minimal cut representations of the binary coherent structure

functions (See Barlow & Proschan, 1975, p. 9), an axiomatic

approach is used by El-Neweihi et al and Griffiths to develop

their models. It turns out that the later developments include

as a special proper subclass of multistate system structures,

the one introduced earlier by Barlow & Wu.

In this paper, we establish verifiable conditions that

characterize the Barlow & Wu models, within a larger class of

multistate system structures which contains all the models in-

troduced by El-Neweihi et al.

3

Let S {O,l,....,m} denote the set of all possible states

of both the system and the components and let C- {l,2,...,n}

be the component set. The vector x= (xI .... ox), in Sn, denotes

the vector of states of each component and we write x < z when-

ever <z i , for i=1,2,...,n. If x< z and x <z i for some

i, we write x< z. Other special notation include:

(k~)-(x I ) ..... x Xi-1 k,X i+l , . . . ,x na ) , keS, i4EC

k - (k,k, ...,k) , keS.

The class of multistate system structures introduced by

El-Neweihi et al is the following:

DEFINITION 2.1. - A function E: Sn - S, is called an EPS multi-

state system structure if:

(2.1) is monotone non-decreasing;

(2.2) E(k)=k, kES;

(2.3) for each ieC and kcS, there exists xeSn, such that

E(kix) -k and (.%,x) k if Z 'k.

Let now P= {Pi: 1<i<p} be a Sperner covering of C,

i.e., a collection of non-empty subsets of C such that Pi P

whenever i o j and UPi = C (When the covering condition is dropped

we shall refer to P simply as a Sperner system on C). It is

easy to see that, • . ***. I.-

(x) - max m n x i , xeSn , . ...... .. t. L

1< j <p ieP

4 , 1 - ,

I r

ohm"

4

defines an EPS multistate system structure, and models of this

form were first suggested by Barlow & Wu, Multistate system

structures of this form will be referred here as BW. Notice

that the class P of subsets of C clearly satisfy the require-

ments for being the min path sets of a binary coherent structure.

A concept analog to that of a min-path vector for an EPS

multistate system structures has been introduced by El-Neweihi

et al which can be used, as in the binary case, to determine

system state.

DEFINITION 2.2 - A vector xeSn is called a connection vector

to level k if *(x) - k. l~rthermore, if *(z) <k, whenever z <x,

we say that x is critical. ///

For k- 1,2,...,m let Ck denote the set of all critical

connection vectors to level k, and for xECk put

Ck(X)- {i:xii .

Then, it is shown in El-Neweihi et al (1978) that for k- 1,2,...,m,

we have u Ck(x)W=C and that (x)> k if and only if x>z forxC.

some zeC 6 nd some Z >k.

3. CHARACTERIZATION OF BW MULTISTATE SYSTEM STRUCTURES

From the reliability point of view, the class of natural

model candidates for multistate system structures consists of

the functions &: Sn S such that

5

(3.1) E is monotone non-decreasing

and

(3.2) t(o) - m- E(m)= 0

and which from now on will bear the name of multistate system

structures (MSS). The class of all MSS's will be denoted by

M.

Note that conditions (3.1) and (3.2) respectively state

that system does not degrade when one or more components up-

grade, and that whenever all the components fail, or work at

best performance levels, the system either fail, or work at

its best performance level, respectively.

For aSS's, the notion of critical connection vector can

be easily extended to provide a corresponding version of the re-

sult mentioned on the last paragraph of section 2.

DEFINITION 3.1 - Let E: Sn - S be an MSS. For k= 1,2,...,m we

say that xeSn is an upper k-vector if E(x)> k. Furthermore, an

upper k-vector is called critical if &(z) < k whenever z < x. The

set of all critical upper k-vectors will be denoted by Pk and

for xePk we shall let

THEOREM 3.2 - Let E:Sn - S be an MSS. Then, for k- 1,2,...,m,

E(x)>k iff x>z for some zcP k .

6

PROOF - Sufficiency is obvious. To prove necessity consider

the procedure of successively decrease the values of each com-

ponent of x, subject to the restriction that the value of

j does not drop below k. Since &(.O) = O, this procedure stops

when we reach a vector zeSn for which &(z) > k and e(w) < k

whenever w< z. Clearly x> z and ZEPk.///

The class M contains as a special subclass, the EPS, and

hence be BW, multistate system structures defined in Section 2.

What we shall do next is study the behavior of elements of M

under property P stated below, and show that this property cap-

tures the axiomatic essence of the BW multistate system struc-

tures when imposed on the suitable subclass of M.

(3.3) PROPERTY P: If k-1,2,...,m and &(x)>k, there exists

z4{O,k}n such that z< x and &(z)> k.

LEMMA 3.3 -Let : Sn - S be an MSS satisfying property P.

Then

&({O,k}n ) - {O,k} , k= 1,2,...,m

iff ({O,m}n ) - (O,m}.

PROOF - The "only if" part is automatic. To prove the "if" state-

ment let us first notice that under property P, we must have

&(x)< k, for all xc(O,k}n and each k- 1,2,...,m.

Since the assertion is now valid for k-rm, assuming that

7

it holds for some k, l< k< m, and letting xE'{O,I-l1n, we have

that y = -- x e (O,k1n so that &(x)-O if (y)O . If onTk (-) --

the other hand e(y) = k,it follows fromproperty P that there

exists ze{O,k-llI such that z< y and &(z)> k-1, and since

~x < y,we must have E(x) > k-i. The observation made in the

beginning of the proof shows that in fact (x) = k-i and the

proof is complete. ///

LEMMA 3.4 - Let &: Sn- S be an MSS satisfying property P and

assume that ~(Omn Om.For k- 1,2,... ,m, we have xEPk

iff .1 XCP1

PROOF -Let xeP .* Since property P holds we have that xe{O,k} n

kk

such that z <x and (z)>l1. Obviously z < -1x, so that x)>1

To show that . -P 1 ntc that if E (w) >_ rsm w < -1x,

there must exist yE{0 ,1 }n such that y< w and E(y)> 1. Then,

ky E{O,k In, E(ky) > E(y)>lI and from Lemma 3.3 we must have

(k y) = kand k y <x , contradicting the f act that xcP k'

The converse is proven daing the same type of arguments. ///

As a consequence of the results of Lemmas 3.3 and 3.4 we

have the following general result regarding MSS's.

THEOREM 3.5 - Let &: S -~ S be an 11SS. Then

(3.4) (x)- max minm- <J<p iepj

for all xes , where {P l< J 1p} is a Sperner system on

|4

8

C = {1,2,... ,n}, iff g({O,m}n) = {O,m} and property P is

satisfied.

PROOF - It is easy to see that if &: Sn - S is of the form

(3.4) then ({O,ml n ) - {O,m} and property P holds.

If, on the other hand, these two conditions are satisfied,

we have

&(x)>k iff x> z for some zEPk (theorem 3.2)

itff x>kw for some weP 1 (lemma 3.4)

iff min xi >k for some weP1i P1 w) 1 - ~

iff max min x > k.

The result now follows if we observe that {P,(W); wePl is a.

Sperner system of subsets of C. 1/1

We remark at this point that the class of functions F: Sn S

satisfying conditions (3.1), (3.3) and

(3.5) &({O,m} n ) = {O,m}

is still larger than the class of BW multistate system structures,

and the reason being is that the Sperner system {Pj: 1< j <p} in

the representation (3.4) of may not cover C.

Nevertheless this covering property can be achieved through the

following notion of component relevance:

(3.6) for each ieC - {1,2,... ,n}, there exists xeS n such that

(O1 ;x) <

9

We thus have the following characterization.

THEOREM 3.6 - The function : Sn - S is a BW multistate system

structure iff (3.1), (3.3), (3.5) and (3.6) hold.

PROOF - Necessity is again obvious. To prove sufficiency all

we have to show, in virtue of Theorem 3.5, is that

U{P(w): =C.

Fixing ieC it follows from (3.6) that there exist xcSn such

that

E(0i,x ) < k < E(,mi,x)

for some k, k= 1,2,..., m. From Theorem 3.2 there exist zEPk such

that z< (mi,x) and obviously zi > 0, since otherwise z< (0i,x)

and E(z) < k (recall that zePk => z{,k }n). Therefore1

iEPk(z) or equivalently ie P1 (.z) by Lemma 3.4, and the proof

is complete.///

In the diagram below we depict the various classes of multi-

state system structures involved in our discussion.

MS- (3.1)+(3.3)+(3.5)

EP-

t BW

...- -7 -

10

4. FINAL REMARKS

1) From the observation made right after Theorem 3.5,

it follows that the class functions satisfying condition (3.1),

(3.3) and (3.5) includes functions &: Sn - S which may be

constant in some of its arguments. In other words, it in-

cludes multistate system structures with "inessential" compo-

nents. Condition (3.6) enters here to require that every com-

ponent be essential in some sense. This condition was first

used by Griffith (1979) to define weakly coherent multistate

system structures. Under property P it follows from this

additional condition that u P1 (y)= C.yEPel

A stronger result can actually be stated:

Proposition 4.1. Let &: Sn _ S is an MSS and kES-{O}. Then

U{Pk(w)'.; wePkI = C

if and only if for every iEC there exists xeSn such that

3(Oi, ) < k< -(mi,x).

The above result suggests a new notion of component relevance

which can be stated as

(4.1) "For every ieC and keS-{O} there exists xESn such that

(0i, < k<

This requires, in some sense, that every component be relevant

for system performance at all levels.

Recall that in (3.6) we introduced a notion of component rele-

vance due to Griffith (1981) in order to characterize a BW MSS;

which is weaker than the one above. However in the presence of

property P, we have

11

Proposition 4.2. Under property P. the notions of component rele-

vance (3.6) and (4.1) are equivalent. ///

2) It is interesting to remark that the class of EPS

multistate system structures for which property P hold does

reduce to the BW ones in the simple case where m= n= 2. However,

as the following example shows, this is not true in general.

EXAMPLE -S= {0,,2}, C= {1,2,3}

x 2(x)

(2,2,2) 2(2,1,2) 2

(1,2,2) 2

(2,2,1) 1

*(2,2,0) 1

(2,0,2) 2

(0,2,2) 2

(2,1,1) 1

(1,2,1) 1

(1,1,2) 1

(2,0,0) 0

*(0,20) 1

(0,0,2) 0

(2,1,0) 1

(2,0,1) 1(1,,2) 1

(1,2,0) 1(0,2,1) 1

(0,1,2) 1(1,1 ,1) 1

(1,0,1) 1

(1,1,0) 1

(0,1,1) 1

(1,0,0) 0(0,1 ,0) 1

(0,0,1) 0

(0,0,0) 0

12

This function verifies the conditions of definition 2.1

and therefore is an EPS multistate system structure. It can

also be checked that property P holds for the function &. How-

ever the two starred points show that (3.5) is not true and that

is not of the B-W type.

3. After this paper had been written, it cameto our know-

ledge that other forms of characterizing BW multistate system

structures were developed by Natvig (1981) and Block & Savits

(1981). Their results however differ from ours in the sense

that no explicit directly verifiable conditions on C are given.

We add however the following additional remarks that relate

our results with the Natvig characterization.

Assume that &: 3n - S is an MSS and that for some keS-{0}

the following property is verified:

(4.2) "If E(x)>k, there exists zE{O,k}n such that z<x

and F(x) >k, i.e. property P introduced in the preceding

section holds just for some level kcS-{O}. It is easy to see that

if keS-{O} is one of the levels for which (4.2) holds and

EP kwe must have xs{O,k}n and consequently

E(x) >k iff max min xi > k,

We pk iEPk( )

which again follows from Theorem 3.2. This can be reworded as

(x) > iff = k Ifor some binary, not necessarily coherent, monotone structure

13

function where ak(x)e (0,I} n has i-th component equal to

1 iff xi > k. Furthermore, it follows from proposition 4.1

that the binary monotone structure function k will be coherent

iff for every ieC there exist xES such that

E(Oi, ) < k < (mi,O)~

We recall that an MSS &: Sn _* S is defined by Natvig to

be a multistate coherent system of type 2 iff there exist binary

conherent monotone structure functions 1,E2.".m such that

(X) > j <-> Ek (a k(x)) =, k-i1,2,...,m.

From the observation made above we have

THEOREM 4.3 - An MSS C: Sn - S is a multistate coherent system

of type 2 iff (3.3) and (3.6) hold.

PROOF - Illows immediately from above observations and proposi-

tion 4.2. ///

As a final remark we add the following more explicit result

that combines our approach with that of Natvig's, whose proof

we omit.

THEOREM 4.4 - A multistate coherent system of type 2, 1: Sn + S

is a BW-MSS iff ((O,m}n) (O,m}.

-7

14

REMtNCES

E1 Barlow, R.E. and Proschan, F. (1975), Statistical Theoryof Reliability and Life Testing, Holt, Rinehart and Winston,New York.

(2] Barlow, R.E. and Wu,A.S.,Coherent Systems with multi-statecomponents. Mathematics of Operations Research, Vol. 3,No. 4, 275-281 (1978).

C3] Block, H. and Savits, T.,"A Decomposition for MultistateMonotone Systems." Research Rept. #81-02. Dept. of Math.and Stat., University of Pittsburgh.

C4) El-Neweihi, E., Proschan, F., Sethurman, J., MultistateCoherent Systems, J. AppI. Prob. 15, 675-688 (1978).

[51 Griffith, W.S., "Multistate R;Sliability Models", ResearchReport No. 78-02, Dept. of Math. and Stat., University ofPittsburgh (1978).

£6) Natvig, B., "Two Suggestions on How to Define a MultistateCoherent System", Research Report; University of Oslo (1981).

ACKNOWLEDGEMENTS

We would like to thank Professor Henry W. Block, while

visiting of the University of Slo Paulo, for reading the first

draft of this paper and making many helpful comments.

.... -- A-

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ON THE AXIOMATIC THEORY OF MULTISTATE Technical - October 1981

COHERENT STRUCTURES 6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR.) II. CONT'RACT OR GRANT 14NMUIER(.)

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~~~~Flavio Wagner Rodrigues ____________

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Ill. SUPPLEMENTARY NOTES

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Multistate system structures, reliability

20 ABSTRACT (Continwe an to-or". side it neOcssay and Identify by black number)

Mathematical models for multistate reliability systems of mul-tistate components have been proposed by Barlow & Wu (1978), ElNeweihi et al (1978) and Griffiths (1981). Unlike the approachused by Barlow & Wu, the other authors preferred to establishtheir classes of models through sets of axioms, all extending theearly binary notions and all containing as special cases the classof models suggested by Barlow & Wu. Since the Barlow & Wu approachis essentially set theoretic, and since in the other two approaches

D 1jFAN"3 1473SECU41RIT I jfRI PAGE (Men Dat. Enered,,

UNMW.ARS FIEDECuRITY CL.ASSIFICATION OF TNIS PAGIE(ffto oe &[0

thede models were not characterized among the lhrger classes, one

question that arises is whether these models can be characterized

by a set of axioms in the same way as their counterparts. In this

paper we do just that and obtain a better understanding of

Barlow & Wu models.

I

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